Isotropy and Anisotropy

For isotropic materials all reflections represent concentric rings (Debye-Scherrer rings) in an image recorded on 2D detector if during exposure the detector was 

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Peaks in Isotropic and Anisotropic Scattering Patterns 8.2.2.1 Isotropy and Anisotropy... 

Isotropy and anisotropy fluid invasion in cross-bedded sands, 153... 

Combination of isotropy and anisotropy Since any orbital may be considered as a hybrid of suitable combinations of s, p or d orbitals, so also may a hyperfine coupling be divided into a contribution arising from p or d orbitals (anisotropic) and s orbitals (isotropic). In general both isotropic and anisotropic hyperfine couplings occur when one or more nuclei with 1 0 are present in the system. The whole interaction is therefore dependent on orientation and must be expressed by the second rank A tensor as given in Equation [13]. The A tensor may be split into an anisotropic and isotropic part as follows... 

Define isotropy and anisotropy with respect to material properties. 

Tensorial character is relevant for rocks with an anisotropy (originated by lamination, preferred direction of fractures, grain axis or mineral orientation, etc.). The terms isotropy and anisotropy refer to the directional dependence of considered (tensorial) properties. Isotropic expresses that a vectorial property has at any point the same magnitude for all directions if the magnitude shows a directional dependence, then the material is anisotropic. 

Accordingly, for a given Raman-active resonance r, the amplitudes of XijM can be expressed in terms of the isotropy and symmetric anisotropy invariants of the corresponding spontaneous Raman scattering tensor, a2 and 7S2, respectively. In the case of frequency-degenerate CRS considered here, the two relevant independent tensor components assume the following form [33, 34] ... 

For most particulate composites the mismatch between the particles and the matrix is more important than the anisotropy of either component (though alumina/aluminium titanate composites provide a notable exception and are described below). The main features of the stresses can therefore be understood in terms of a simple elastic model assuming thermoelastic isotropy and consisting of a spherical particle in a concentric spherical shell of matrix with dimensions chosen to give the appropriate volume fractions. The particles are predicted to be under a uniform hydrostatic stress, ap after cooling. If the particles have a larger thermal expansion coefficient than the matrix, this stress is tensile, and vice versa. For small particle volume fractions the stress... 

The traditional definitions of the three principal value components (5i 1, 22, and 533), isotropy 5iso, anisotropy and asymmetry of the chemical shift are equivalent to those of the magnetic shielding. This traditional scheme is indefinite. It leads to a loss of physical and chemical generality which is disadvantageous. Therefore, besides the isotropy two new parameters, span Q and skew k, are often introduced and used instead of the anisotropy A3 and... 

Regarding the macromechanics perspective, assuming the material to be almost uniform, its anisotropy can be exploited as an advantage. The average behavior of the material can be predicted and controlled by knowing the properties of its constituents. Nevertheless, the anisotropic analysis is more complex and more dependent on the measurement procedures, whereas the analysis of conventional materials is easier due to their isotropy and uniformity. 

Layer-induced anisotropy (transversely isotropy) and its consequences for the interpretation of seismic data are the subjects of papers by Helbig (1958, 1992), Backus (1962), White (1983), Thomsen (1986, 1995), Schoenberg and Muir (1989), and Schoenberg (1994). 

This automatic velocity measurement method (AUTO-V) has also been suecessfijlly applied to other materials sueh as non-ferrous metals and even eeramics and plasties. The only requirement for this type of sequential velocity and thickness measurement is a certain isotropy of the material materials having anisotropie properties will lead to incorrect thickness readings due to the velocity difference in the two orthogonal directions. 

Phase-averaged values of 4 in a plane midway between two baffles of a stirred tank have been plotted in Fig. 1 (from Hartmann et al., 2004a) for two different SGS models (Smagorinsky and Voke, respectively) in LES carried out in a LB approach. The highest values, i.e., the strongest deviations from isotropy, occur in the impeller zone, in the boundary layers along wall and bottom of the tank, and at the separation points at the vessel wall from which the anisotropy is advected into the bulk flow. In the recirculation loops, the turbulent flow is more or less isotropic. 

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